(Original post by SimonM)
No it's not. Hom(R^2,R^2) is the set of linear maps from R^2 to R^2, which can easily be identified with 2x2 matrices (which is isomorphic to R^4 as a real vector space)

yea thats what i thought, someone said it was isomorphic to r^2 but as i was posting it i thought you'd need the matrices (1 0, 0 0) ( 0 1, 0 0) and so on to be able to create any matrix which is a linear map from r^2 to r^2 thanks for clarifying that

(Original post by SimonM)
No it's not. Hom(R^2,R^2) is the set of linear maps from R^2 to R^2, which can easily be identified with 2x2 matrices (which is isomorphic to R^4 as a real vector space)

okay quick question, theres a one to one correspondence with K^2x2 for "a given basis", does this matter? if its isomorphic we're assuming its injective but by the definition of hom do we choose a given basis? cause if we can consider all basis then i'm starting to doubt in my head its isomorphic

but i'm assuming its because we only consider a fixed basis in R^2 which gets mapped to something else?

basically am i right in thinking Hom(R^2,R^2) = set of transformations with respect to a given basis of R^2 which are mapped to R^2?

(Original post by Chaoslord)
okay quick question, theres a one to one correspondence with K^2x2 for "a given basis", does this matter? if its isomorphic we're assuming its injective but by the definition of hom do we choose a given basis? cause if we can consider all basis then i'm starting to doubt in my head its isomorphic

but i'm assuming its because we only consider a fixed basis in R^2 which gets mapped to something else?

basically am i right in thinking Hom(R^2,R^2) = set of transformations with respect to a given basis of R^2 which are mapped to R^2?

Hom(V, W) is independent of basis, but (just like any vector space) can be given a basis itself, and any vector space U over F is isomorphic to F^n for some n, with isomorphism determined by choosing a basis.

(Original post by generalebriety)
Hom(V, W) is independent of basis, but (just like any vector space) can be given a basis itself, and any vector space U over F is isomorphic to F^n for some n, with isomorphism determined by choosing a basis.